The following question is from Kirillov's Introduction to Lie Groups and Lie Algebras, and my attempt is the following: $$\sum_{n\geq 0}\frac{(t\partial_x)^n}{n!}f=(\exp{t\partial_x})f=\exp(\rho(t))f=\rho(\exp(t))f$$ The last identity comes from the fact that $\exp\circ\rho_*=\rho\circ\exp$, where $\rho_*$ is an induced morphism of $\rho$. (The author used the same symbol $\rho$ for both of these morphisms.) But the last expression seems to have no connection with what I want to show, which is $f(x+t)$. What should I do now?
